1. Field of the Invention
The present invention relates to numerically analyzing a behavior of a continuum, such as a fluid, using a particle method. More particularly, it relates to accurately determining a particle positioned on a free surface among particles that approximately represent the continuum, such as the fluid.
2. Related Art
A particle method is a technique whereby a continuum is approximately represented by a group of a plurality of particles, an algebraic relational expression is defined between these particles so that the group of particles imitates a behavior of the continuum, and analysis is performed by numerical calculation. Put more generally, the continuum subjected to calculation analysis is collectively modeled with a free surface by the plurality of particles. The particle method has the advantages that mesh generation is unnecessary and also that even a large deformation of a fluid subjected to analysis can be easily handled.
Known particle methods for fluids include the SPH (Smoothed Particle Hydrodynamics) method that calculates a behavior of a compressible fluid by an explicit solution technique, the MPS (Moving Particle Semi-implicit) and the ISPH methods that calculate a behavior of an incompressible fluid by a semi-implicit solution technique. In a calculation process of a particle method for an incompressible fluid, a pressure of the fluid is calculated as a solution of Poisson's equation defined so as to reflect an incompressibility condition.
However, an unnatural disturbance included in the actually calculated pressure poses a significant problem for practical use of the particle method.
In order to calculate the pressure of the fluid as the solution of Poisson's equation, there is a need to determine whether or not a particle approximately representing the fluid is positioned on the free surface. Each particle determined as being positioned on the free surface undergoes a process of fixing the pressure at zero, which corresponds to a process of imposing a boundary condition on Poisson's equation.
However, the solution of Poisson's equation significantly varies depending on a difference in boundary condition. Accordingly, if the determination of whether or not a particle approximately representing the fluid is positioned on the free surface cannot be made accurately, an unnatural disturbance is caused in a pressure distribution calculated in the particle method.
In a most representative method for determining whether or not a particle (i) is positioned on the free surface, the determination is made according to whether or not a “particle number density” ni of the particle (i) satisfies the following expression (1) with reference to a fixed value n0:ni<βn0  (1)
The particle number density ni is defined as a total sum of weights defined for other particles that exist within a predetermined range (re) with respect to the particle (i). n0 is a reference particle number density, and β is a constant less than 1 specified according to the model.
When incompressibility is strictly satisfied in the calculation process, a particle number density of a particle existing in the fluid matches the reference particle number density n0. This determination method is based on the ground that, since no particle is positioned outside the fluid, a smaller number of particles influence the weight total sum calculation for the particle positioned on the free surface, resulting in a lower particle number density of the particle.
There are several other known threshold-based determination methods formulated based on the substantially same ground.
However, in such a threshold-based determination method, there is an instance where a particle (a particle inside the analysis object) other than the particle positioned on the free surface is erroneously determined as a free surface particle. This is largely because, in the actual calculation of the particle method, the particle number density of each particle positioned inside the fluid is not strictly held constant but has a deviation.
In particular, since in more than a few cases the particle number density of the particle positioned inside the fluid is lower than the particle number density of the particle that is supposed to be determined as being positioned on the free surface, there is a substantial difficulty in using this method alone.
A method for simultaneously taking particle distribution asymmetry into consideration rather than singly applying the threshold determination for the particle number density is also known. In this method, a numerical value that serves as a criterion of asymmetry in a distribution of other particles within a predetermined range with respect to one particle is defined, and a particle that is determined as asymmetrical as a result of the threshold-based determination using the numerical value and also determined as having a low particle number density is determined as a particle positioned on the free surface.
This determination method is based on the ground that, since no particle is positioned outside the fluid, not only the particle positioned on the free surface has a lower particle number density but also there is a deviation in a distribution of other particles around the particle. This method is publicly known as disclosed in International Patent Publication No. WO/2010/032656.
However, even in the case of using this determination method that takes asymmetry into consideration, the particle positioned on the free surface cannot always be accurately specified. For example, merely by rotating about one particle by an arbitrary angle while maintaining relative positional relationships of all particles and fixing a coordinate system, the numerical value used as the criterion of asymmetry for the particle changes.
This leads to such a situation where different determination results are obtained depending on circumstances when the same determination result should be obtained. Thus, it is difficult to set the threshold for the determination.
Besides, the determination result is significantly affected when the parameter used for the determination is changed, and in more than a few cases the determination method that takes asymmetry into consideration fails to effectively function per se.
Typically, in the case where a particle positioned inside the fluid is erroneously determined as being positioned on the free surface, a larger disturbance is caused in the calculated pressure distribution when the particle is positioned farther inside the fluid.
It can be understood from the above description that, as long as the conventional technique is used, it is impossible to fundamentally prevent such erroneous determination that causes a significant disturbance in the pressure distribution while properly determining the particle positioned on the free surface, even though the parameter such as the threshold is carefully set. Especially in the case of performing analysis by large-scale calculation using a large number of particles, a more serious situation arises, making it extremely difficult to realize high-accuracy analysis which is an intended purpose of large-scale calculation.
As described above, it is extremely difficult to accurately determine whether or not a particle is positioned on the free surface. This problem, however, is unavoidable when performing analysis that uses the particle method.
The practical use of the particle method is expected to be greatly facilitated if such a free surface determination method can be established that, while characterizing a particle positioned on the free surface by a simple index as in the conventional technique, can fundamentally prevent erroneous determination that a sufficiently inside particle is a free surface particle, which causes a particularly significant disturbance in pressure.
Japanese Unexamined Patent Publication No. H7-334484—mandatory publication at 18 months after filing date in Japan, but which has not been published yet is referenced herein.